泰勒 1/(1-x),14

解答

泰勒

\frac{1}{1 - x}, 14

- \frac{1}{13} + \frac{1}{169} (x - 14) - \frac{1}{2197} (x - 14)^{2} + \frac{1}{28561} (x - 14)^{3} - \frac{1}{371293} (x - 14)^{4} + \dots

求解步骤

应用泰勒公式

= - \frac{1}{13} + \frac{\frac{d}{d x} ( \frac{1}{1 - x} ) ( 14 )}{1 !} (x - 14) + \frac{\frac{d ^{2}}{d x ^{2}} ( \frac{1}{1 - x} ) ( 14 )}{2 !} (x - 14)^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( \frac{1}{1 - x} ) ( 14 )}{3 !} (x - 14)^{3} + \dots

求导数

= - \frac{1}{13} + \frac{\frac{1}{169}}{1 !} (x - 14) + \frac{- \frac{2}{2197}}{2 !} (x - 14)^{2} + \frac{\frac{6}{28561}}{3 !} (x - 14)^{3} + \frac{- \frac{24}{371293}}{4 !} (x - 14)^{4} + \dots

整理后得

= - \frac{1}{13} + \frac{1}{169} (x - 14) - \frac{1}{2197} (x - 14)^{2} + \frac{1}{28561} (x - 14)^{3} - \frac{1}{371293} (x - 14)^{4} + \dots